Demodulation of multiple-carrier phase-modulated signals

ABSTRACT

A Fourier transform is applied ( 205 ) to a received multiple-carrier phase-modulated signal. The output of the transform includes real and imaginary elements that are associated with the multiple carriers. These elements are processed to provide a magnitude ( 209 ) and a sign ( 211 ) for the quadrature, Q, and in-phase, I, components of the desired signal. Q and I are processed ( 213 ) to yield the desired output signal that approximates the originally obtained signals that were phase-modulated and transmitted on the multiple carriers.

FIELD OF THE INVENTION

[0001] This invention relates to communications, including but notlimited to demodulation of multiple-carrier phase-modulated signals.

BACKGROUND OF THE INVENTION

[0002] Fiber optic sensor systems utilize phase-modulated carriers totransmit information of interest in the phase of an optical signal. Thecarrier is manifested as a sinusoidal phase modulation of the opticalwave that is used by a pressure sensitive optical sensor. The pressureinformation transduced by the optical sensor adds an additional phasemodulation to the optical signal. When the optical signal is received ata remote location, usually via fiber optic media, the pressureinformation is extracted from the optical signal in a process known asdemodulation.

[0003] Demodulation involves converting the optical signal to anelectrical signal. In digitally oriented systems, the analog electricalsignal is passed through an analog-to-digital (A/D) converter, afterwhich the desired pressure information may be extracted via digitalmeans. In a Frequency Division Multiplexed (FDM) system, multipleoptical carriers are combined through an array of sensors. The resultantelectrical signal is very complex. This situation is analogous to anFrequency Modulation (FM) cable system where many carriers or channelsare transmitted on a single conductor. When demodulating an FM signal,the conventional process is called heterodyning.

[0004] While similar to heterodyning, the conventional process ofdemodulating an optical sensor system is called homodyning. The processof utilizing homodyning to extract a signal with many carriers is veryprocess intensive. Two mixers are utilized for each of the multiplexedcarriers. These mixers down-convert the carrier signal's first andsecond harmonics to baseband. The mixing process produces numerousundesirable harmonics that must be filtered out. A series of lowpassfiltering stages is utilized to recover the desired in-phase andquadrature signal. When such a process is performed on carrier signalsin the MHz range, the processor performing the mixing and filtering maybe required to perform billions of mathematical operations per second.Although such a process is not impossible, it is difficult to implementsuch a process with hardware that fits in a small space and consumes lowpower.

[0005] Accordingly, there is a need for an apparatus for and method ofdemodulating multiple-carrier phase-modulated signals that does notproduce unwanted signals while utilizing a reduced amount of computationwhen compared to known methods.

SUMMARY OF THE INVENTION

[0006] A method of demodulating multiple-carrier phase-modulated signalscomprises the steps of receiving a multiple-carrier phase-modulatedsignal and converting the multiple-carrier phase-modulated signal fromanalog to digital, yielding a digital signal. A Fourier transform isperformed on the digital signal, yielding a plurality of real andimaginary elements associated with an n^(th) carrier, where n is aninteger. A magnitude of an in-phase component for the n^(th) carrier anda magnitude of a quadrature component for the n^(th) carrier isdetermined from the plurality of real and imaginary elements associatedwith the n^(th) carrier. A sign is established for the in-phasecomponent for the n^(th) carrier from a first subset of the plurality ofreal and imaginary elements associated with the n^(th) carrier. A signis established for the quadrature component for the n^(th) carrier froma second subset of the plurality of real and imaginary elementsassociated with the n^(th) carrier. The in-phase component and thequadrature component are processed to yield an output signal associatedwith the n^(th) carrier.

BRIEF DESCRIPTION OF THE DRAWINGS

[0007]FIG. 1 is a block diagram of a communication system in accordancewith the invention.

[0008]FIG. 2 is a flowchart showing a method of demodulatingmultiple-carrier phase-modulated signals in accordance with theinvention.

[0009]FIG. 3 is a diagram illustrating multiple frequency carriers inaccordance with the invention.

DESCRIPTION OF A PREFERRED EMBODIMENT

[0010] The following describes an apparatus for and method of applying aFourier transform to a received multiple-carrier phase-modulated signal.The output of the transform includes real and imaginary elements thatare associated with the multiple carriers. These elements are processedto provide a magnitude and a sign for the quadrature, Q, and in-phase,I, components of the desired signal. Q and I are processed to yield thedesired output signal that approximates the originally obtained signalsthat were phase-modulated and transmitted on the multiple carriers.

[0011] A block diagram of a communication system is shown in FIG. 1.Although the system of FIG. 1 is depicted as an optical communicationsystem, the principles of the invention may be successfully applied toradio frequency (RF) communication systems or other types of systemsthat employ multiple-carrier phase-modulated signals. A multiple-carriergenerator 101 generates a plurality of carriers, typically one for eachof a plurality of sensors in a sensor array 103. Each of the sensors inthe sensor array 103 receives or senses data that the sensor thenproportionally modulates onto one of the carriers. The sensors may beoptical sensors that receive data and generate an optical output signal.The outputs from the sensors in the sensor array 103 are input to atransmitter 105. For the optical example, the transmitter 105 combinesthe sensor array's 103 signals and couples them together fortransmission through a communication medium 107, which for optical data,is an optical medium 107 such as fiber optic cable or air or empty spacein the case where the transmitter is a laser.

[0012] The present invention is applicable to other transmission devicesand transmission media, such as RF, wireline, acoustic, and so forth,whether solid, liquid, or gas. In the case where an RF signal is to betransmitted, the transmitter 105 is an RF transmitter that receives andtransmits over the air multiple-carrier phase-modulated signals that aregenerated by devices known in the RF art. The transmitter 105 providesappropriate amplification to transmit the RF signal in appropriatecommunication channels 107, such frequencies and/or timeslots. Similaradjustments are made for other transmission types.

[0013] The multiple-carrier phase-modulated signal is received at ademodulator 109 that demodulates the signal. The demodulator 109 mayinclude a polarization diversity detector (PDD) 111 that converts anoptical signal to an electrical signal, as necessary. An optionalamplification and anti-aliasing filter 113 provides any necessaryamplification/attenuation or anti-aliasing functions. Ananalog-to-digital (A/D) converter 115 converts the signal from an analogsignal to a digital signal, as known in the art. Multiple A/D converters115 may be implemented. A fast Fourier transformer 117 takes a fastFourier transform (FFT), which is a fast algorithm version of a discreteFourier transform (DFT), as known in the art. Alternatively, a DFT maybe used instead of an FFT.

[0014] The output of the FFT 117 is input to a frequency bin selector119 that takes the FFT output data and places it in frequency binsassociated with each carrier's first harmonic and second harmonic. Afrequency bin is a spectral frequency corresponding to the complexnumber output by the FFT for that frequency. An FFT outputs as manyfrequency bins as is the size of the FFT. For example, a 256-point FFToutputs 256 complex numbers corresponding to 256 spectral values in 256frequency bins.

[0015] A magnitude block 121 (magnitude determiner) determines themagnitude of the in-phase (I) and quadrature (Q) signal components foreach carrier. The in-phase (I) signal value for a carrier is determinedfrom the frequency bin data associated with the 2nd harmonic of thecarrier, and the quadrature (Q) signal is determined from the frequencybin data associated with the 1st harmonic of the carrier. A sign block123 establishes a sign for the I and Q signal components for eachcarrier. The sign block 123 (an in-phase sign establisher and aquadrature sign establisher) may be a single block or separate blocksthat establish a sign for the I and Q. The calibration path 125 receivesthe FFT output and optionally performs various calibration functionsthat may be useful in the sign determination process that takes place inthe sign block 123. Further details on the functionality of thefrequency bin selector 119, magnitude block 121, sign block 123, andcalibration path 125 are described below. A process that takes anarctangent 127 of the Q/I quotient yields the desired recovered signal.Additional processing on the received signal may also be performed.

[0016] Because it is desired to recover the signal Φ_(n)(t) for then^(th) carrier from the received multiple-carrier phase-modulatedsignal, the following is provided to show how to obtain Q_(n)(t) andI_(n)(t) in order to determine Φ_(n)(t) for the n^(th) carrier. Avoltage that represents an optical signal may be written as:

V=A+B cos(C cos ωt+Φ(t)),  (1)

[0017] where:

[0018] V=voltage of the signal;

[0019] A=DC (direct current) offset component of the voltage;

[0020] B=peak amplitude of the time varying portion of the voltage;

[0021] C=modulation depth of the phase generated carrier;

[0022] ω=modulation frequency;

[0023] t=time; and

[0024] Φ(t)=signal of interest to be recovered.

[0025] In a frequency division multiplexed (FDM) system, multiplecarrier signals are present, and each carrier signal simultaneouslyobeys the above equation. Applying equation (1) to a multi-carriersystem yields:

V _(n) =A _(n) +B _(n) cos(C _(n) cos ω_(n) t+Φ _(n)(t))  (2)

[0026] where:

[0027] V_(n)=voltage of the n^(th) carrier signal;

[0028] A_(n)=DC offset component of the n^(th) carrier voltage;

[0029] B_(n)=peak amplitude of the time varying portion of the n^(th)carrier voltage;

[0030] C_(n)=modulation depth of the n^(th) phase generated carrier;

[0031] ω_(n)=modulation frequency of the n^(th) carrier;

[0032] t=time; and

[0033] Φ_(n)(t)=signal of interest on then^(th) carrier to be recovered.

[0034] The combined signal, as present on a single communication medium,such as a conductor, is: $\begin{matrix}{{S = {\sum\limits_{n = 1}^{N}\quad V_{n}}},} & (3)\end{matrix}$

[0035] where:

[0036] S=combined signal of all the carriers;

[0037] N=total number of carriers; and

[0038] V_(n)=induced voltage of the n^(th) carrier.

[0039] Equation (2) may be rewritten using Bessel functions as:$\begin{matrix}{{V_{n} = {A_{n} + {B_{n}\left\{ {{\left( {{J_{0}\left( C_{n} \right)} + {2{\sum\limits_{k = 1}^{\infty}\quad {\left( {- 1} \right)^{k}{J_{2k}\left( C_{n} \right)}\cos \quad 2k\quad \omega_{n}t}}}} \right)\cos \quad {\Phi_{n}(t)}} - {\left( {2{\sum\limits_{k = 0}^{\infty}\quad {\left( {- 1} \right)^{k}{J_{{2k} + 1}\left( C_{n} \right)}{\cos \left( {{2k} + 1} \right)}\quad \omega_{n}t}}} \right)\sin \quad {\Phi_{n}(t)}}} \right\}}}},} & (4)\end{matrix}$

[0040] where:

[0041] V_(n)=voltage of the n^(th) carrier signal;

[0042] A_(n)=DC offset component of the n^(th) carrier voltage;

[0043] B_(n)=peak amplitude of the time varying portion of the n^(th)carrier voltage;

[0044] C_(n)=modulation depth of the n^(th) phase generated carrier;

[0045] ω_(n)=modulation frequency of the n^(th) carrier;

[0046] t=time;

[0047] Φ_(n)(t)=signal of interest on the n^(th) carrier to berecovered; and

[0048] J_(k)=Bessel Function of the First Kind of the k^(th) order.

[0049] Extracting cos Φ_(n)(t) and sin Φ_(n)(t) from equation (4) helpsto obtain Φ_(n)(t), the signal of interest. By applying an arctangentfunction to the sine and cosine terms, the desired signal, Φ_(n)(t), isrecoverable:

Φ_(n)(t)=arc tan(sin Φ_(n)(t)/cos Φ_(n)(t)).  (5)

[0050] The cosine and sine terms of equation (5) are typically referredto as the in-phase and quadrature terms, I and Q, respectively. Thus,I_(n)(t)=cos Φ_(n)(t) and Q_(n)(t)=sin Φ_(n)(t). Substituting for theseidentities in equation (5) yields:

Φ_(n)(t)=arc tan(Q _(n)(t)/I _(n)(t)).  (6)

[0051] Thus, finding Q_(n)(t) and I_(n)(t) and taking the arctangent oftheir quotient yields the desired signal, Φ_(n)(t), for the n^(th)carrier.

[0052] Equation (4) shows that the in-phase term is multiplied by thesummation of the even harmonics of ω_(n) and the quadrature term ismultiplied by the summation of the odd harmonics of ω_(n) and theassociated Bessel functions. Because the summations in equation (4) goto infinity, many terms may be used to extract the desired I and Qterms. Typically, the first harmonic of ω_(n) (that is, ω_(n) itself)and its second harmonic, 2ω_(n), are used. To isolate the terms ofinterest, the remaining terms of equation (4) are typically eliminated.

[0053] In order to have an even balance of power between the in-phaseand quadrature terms, a value for C_(n) is typically chosen such thatJ₂(C_(n)) and J₁(C_(n)) are equal. A value that is suitable for thisaspect is C_(n)=2.62987 for all n. The actual function value ofJ₂(C_(n)) and J₁(C_(n)) are not critical as long as they are equal,because the arctangent takes a ratio of I and Q, and the actual valuesof J₂(C_(n)) and J₁(C_(n)) will cancel out.

[0054] By managing C_(n), commonly referred to as the modulation depth,and eliminating the unwanted terms of equation (4), including the DC (or0 frequency) terms, the in-phase and quadrature terms may be recovered.Appropriate management of the modulation depth directly impacts what isreferred to as the balance of I and Q. The average magnitude of I and Qover time are preferably the same, otherwise the arctangent function ofequation (5) above may not produce the correct result. Techniques tomanage the balance of I and Q for the homodyne approach and the FFTapproach are similar. Likewise, the use of the arctangent function andother subsequent processing necessary to produce the final desired dataafter I and Q are obtained is the same with either approach. Obtaining Iand Q from equation (4) is, however, different using the FFT approach inan FDM system rather than a homodyne approach.

[0055] The homodyne and FFT approaches typically start with an analogfront end filter that removes most harmonics above the 2^(nd) harmonicof the highest carrier frequency in the system. The homodyne systemperforms a real mix of each of the carriers and its 2^(nd) harmonic tospectrally move the result to DC, where the I and Q signals may beextracted, as is similar to the process performed in a conventionalheterodyning radio system. A real mix (a real mix performsmultiplication with real numbers as opposed to a complex mix thatperforms multiplication with complex numbers) of a signal to spectrallymove it down to DC also produces a sum term that creates an unwantedsignal at twice the frequency. These unwanted sum terms are generallyfiltered out with a lowpass filter. A homodyned FDM system requires twomixers and two sets of low pass filters for each carrier that is presenton a common conductor. Although it is typically performed in a digitalenvironment, the homodyne approach may be performed in an analogenvironment. Both the digital homodyne and FFT approaches typicallyutilize analog to digital converters in their receivers. The FFTapproach, however, does not perform mixing operations, thus unwanted sumterms that need to be filtered out are not created. The carriers andtheir 2^(nd) harmonics are used without first spectrally shifting themto DC.

[0056] A flowchart showing a method of demodulating a multiple-carrierphase-modulated signal is shown in FIG. 2. At step 201, amultiple-carrier phase-modulated signal is received and sampled. Thedemodulation process may include, for example, applying a PDD 111,amplifying the received signal, applying an anti-aliasing filter 113,and/or employing an analog-to-digital (A/D) converter 115. In theexample that is described herein, the data is digital data output by theA/D converter 115 as may be represented by equation (3). At step 203,the digital data is collected until M samples are obtained. M is aninteger whose value will be described later.

[0057] At step 205, a Fourier transform of size M is performed. TheFourier transform may be a DFT or FFT. For computational efficiency, anFFT is utilized, as is well known in the art. An example of an FFTprocess is described in a book titled “The Fast Fourier Transform” by E.Oran Brigham, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. The FFTis a computationally efficient way to compute all the possible outputsof equation (7) below simultaneously without the brute force evaluationof equation (7) as it is written. The FFT results are the same as theDFT results. The output of the DFT/FFT process is a plurality of realand imaginary elements that are associated with each of the n carriers.The details of an example of this process are as follows.

[0058] An example of a transform from the time domain to the frequencydomain is: $\begin{matrix}{{{G\left( {h/{MT}} \right)} = {\sum\limits_{p = 0}^{M - 1}\quad {{g({pT})}^{{- {j2\pi}}\quad h\quad {p/M}}}}},} & (7)\end{matrix}$

[0059] where:

[0060] M=number of input samples in the time domain to be used tocalculate the DFT/FFT, commonly referred to as the size of the DFT/FFT;

[0061] p=sample index; an integer between 0 and M−1 relative to thefirst input sample, denoted as sample 0;

[0062] j={square root}{square root over (−1)};

[0063] T=period of time between input samples;

[0064] g(pT)=p^(th) time domain sample at time pT (may be complex orreal);

[0065] h=output frequency bin index and is an integer between 0 and M−1;

[0066] h/MT=frequency represented by the h^(th) bin; and

[0067] G(h/MT)=complex value that represents the complex phasor in theh^(th) frequency bin.

[0068] The sample rate frequency, F_(s), is often used instead of T, andis represented by:

F _(s)=1.0/T.  (8)

[0069] If T is in seconds, F_(s) is in Hz.

[0070] For optimized results with the FFT, e.g., minimal calculationswithout resultant excess noise terms that need to be filtered out, M isselected (or the carriers may be selected) such that each carrier'sfrequency, as well as the frequency of each carrier's second harmonic,matches the frequency of an FFT output bin. For this optimized result,each ω_(n) from equation (4) is expressed as:

ω_(n)=2π(h _(n) /M)F _(s),  (9)

[0071] where h_(n) is an integer such that 0≦h_(n)≦(M/2)−1.

[0072] Additional benefit is gained when the second harmonic of ω_(n),i.e., 2ω_(n), also falls within a bin and equation (9) is met, i.e.,2ω_(n)=2π(2h_(n)/M)F_(s). In other words, when all carrier frequenciesand their second harmonics are integer multiples of 2π(1.0/M)F_(s),which is the frequency resolution of the output of the DFT or the binwidth in frequency, the DFT/FFT calculations are optimized.

[0073] A diagram illustrating an example of a selection of frequencycarriers is shown in FIG. 3. This example shows three frequencycarriers, although a large number of carriers and various differentbandwidths may be utilized, given frequency availability. The examplefrequency diagram shows three carrier frequencies, f₁=3.0 MHz, f₂=3.4MHz, and f₃=3.8 MHz. In this example, the bandwidth for each channel is400 kHz for each frequency. The second harmonics for the carriers f₁,f₂, and f₃ are also shown in the diagram as 2f₁=6.0 MHz, 2f₂=6.8 MHz,and f₃=7.6 MHz, respectively.

[0074] Because the input to the DFT/FFT is real, all imaginarycomponents are zero, thus h_(n) is restricted to half the range of thetotal available bins. For processing speed purposes, a single complexFFT may be used to compute two real DFT/FFTs simultaneously. In thissituation, a complex input stream is formed by assigning one part of theDFT/FFT input as the real part of the complex input stream and anotherpart of the DFT/FFT input as the imaginary part of the complex inputstream, and performing a single complex DFT/FFT. When multiple A/Dconverters are utilized, the outputs of two A/D converters may beprocessed by a single DFT/FFT. When the DFT/FFT is completed, a simplemanipulation of the outputs provides a result that is mathematically theequivalent of performing two real DFT/FFTs.

[0075] Also at step 205, the real and imaginary elements that are outputby the FFT for each carrier and its 2^(nd) harmonic (as well as otherharmonics that were not previously filtered out, for example, by theanti-aliasing filter 113) are separated into frequency bins based oncarrier frequency. Each utilized frequency bin typically includes FFToutput data in the form of a single complex number that has a realelement and an imaginary element associated with the carrier frequencyassociated with that bin. Table 1 below shows an example of frequencybins containing FFT output data in the form of real and imaginaryelements for an 8-carrier system. The example includes 16 bins, each binhaving a real and an imaginary part. Two bins are associated with eachcarrier frequency, one bin containing data corresponding to the 1^(st)harmonic of the carrier frequency (containing data corresponding to Q)and a second bin containing data corresponding to the 2^(nd) harmonic ofthe carrier frequency (containing data corresponding to I). TABLE 1Q(1^(st) harmonic) I(2^(nd) harmonic) a (real) b (imaginary) c (real) d(imaginary) f₁ 1.5 −7.8 7.88 7.43 f₂ −2.5 9.2 −8.34 1.1125 f₃ 7.2 −0.116.21 −4.78 f₄ −4.1 8.55 −3.77 9.432 f₅ 3.7 3.33 −9.05 6.771 f₆ −8.125−7.21 −8.11 6.334 f₇ 5.0978 8.78 5.77 −0.005 f₈ 3.45 −9.88 7.654 −2.3974

[0076] The complex number that is output in each frequency bin is oftenreferred to as a phasor. The complex number is a rotating phasor,because each time the FFT input is stepped M samples forward in time inthe input stream, the next computation of the FFT shows that the phasorhas rotated. When the carriers follow equation (9), it may be shown thatthe rotation is always an integer multiple of 2π. That is, the carrierphase angle rotation from one FFT to the next is:

ω_(n) MT=2π(h _(n) /M)F _(s) MT=2πh _(n).  (10)

[0077] When the FFT is stepped exactly M samples each cycle, the carrierphasor is stationary, thus the carrier is effectively eliminated. Thehomodyne approach utilizes mixers to bring each carrier to DC, whichmixers produce unwanted harmonics that need to be filtered out. Thus,the DFT/FFT approach eliminates the need for mixers and filtering toremove unwanted harmonics, in addition to being computationallyefficient.

[0078] Carriers are typically processed one at a time. At step 207, thefrequency bins for the 1^(st) and 2^(nd) harmonics for the carrier beingprocessed are selected. At step 209, a magnitude of I_(n)(t) and amagnitude of Q_(n)(t) are determined. The magnitude of the complexnumber in the frequency bin associated with the 1^(st) harmonic of then^(th) frequency carrier is directly proportional to the magnitude ofQ_(n)(t). The magnitude of the complex number in the frequency binassociated with the 2^(nd) harmonic of the n^(th) frequency carrier isdirectly proportional to the magnitude of I_(n)(t). The DFT/FFT approachhas the additional benefit that any DC components in the original signalare removed, because DC components show up in the zeroth frequency bin,which is discarded. The following description provides detailsdescribing an example of a method for finding the magnitude of I_(n)(t)and the magnitude of Q_(n)(t).

[0079] Discrete time sample notation common to digital signal processinganalysis will be utilized. In addition, both the phase of the carriersignals and the start sample index of the FFT input data set relative tothe carrier nominal zero phase will be shown. Substituting in equation(4):

ω_(n)

t

ω _(n)(p+L)T+Ψ _(n),  (11)

[0080] where:

[0081] p=an integer representing the pth sample index relative to thestart of process time, 0≦p≦∞;

[0082] L=relative offset of samples to the start of the FFT input, i.e.,the start sample index;

[0083] T=sampling period, which may be the same as in equation (7); and

[0084] Ψ_(n)=relative phase of the n^(th) carrier to time zero.

[0085] After the above substitution in equation (4) and subsequentlyeliminating the Bessel terms of the equation and B_(n) because theseterms will cancel out in the arctangent calculation, the in-phase andquadrature terms in sample notation and in normalized form may bewritten as follows:

Q _(np)=cos[ω_(n)(p+L)T+Ψ _(n)]·sin Φ_(n)((p+L)T)  (12)

[0086] and

I _(np)=cos(2[ω_(n)(p+L)T+Ψ _(n)])·cos Φ_(n)((p+L)T)  (13)

[0087] After applying trigonometric identities and algebraicmanipulation, equations (12) and (13) become:

Q _(np)={cos(ω_(n) pT)cos(ω_(n) LT+Ψ _(n))−sin(ω_(n) pT)sin(ω_(n) LT+Ψ_(n))}·sin Φ_(n)((p+L)T)  (14)

[0088] and

I _(np)={cos(2ω_(n) pT)cos(2(ω_(n) LT+Ψ _(n)))−sin(2ω_(n) pT)sin(2(ω_(n)LT+Ψ _(n)))}·cos Φ_(n)((p+L)T).  (15)

[0089] Q and I as defined in equations (14) and (15) are real numbers.The output of the FFT in a given frequency bin is a complex number. Therelationship between the complex FFT output and the real result ofequations (14) and (15) is as follows. The magnitude of the complexnumber in a frequency bin is proportional to the magnitude of either Qor I, and because (14) and (15) may be positive or negative, the correctsign needs to be established for Q and I. By expanding the exponentialterm of equation (7) and applying Euler's formula:

e ^(−j2πhp/M)=cos(2πhp/M)−j sin(2πhp/M).  (16)

[0090] Because the DFT/FFT is a sum of products involving complexnumbers when the input is real, anything associated with a sine function(with zero phase) correlates with the imaginary part of equation (16)and anything associated with a cosine function (with zero phase)correlates with the real part of (16). As stated earlier, the FFTremoves the carrier parts of equations (14) and (15), which isequivalent to the elimination of the cos(ω_(n)pT), sin(ω_(n)pT),cos(2ω_(n)pT) and sin(2ω_(n)pT) terms. Thus, the real and imaginaryparts of the output of the FFT for the frequency bins of interest (thatis the bins corresponding to the first and second harmonics) areproportional to the remaining corresponding cosine and sine parts ofequations (14) and (15). In transform notation, where

designates “is the transform pair of”:

Re{Q _(n)}

cos(ω_(n) LT+Ψ _(n))·sin Φ_(n)((p+L)T);  (17)

Im{Q _(n)}

−sin(ω_(n) LT +Ψ _(n))·sin Φ_(n)((p+L)T);  (18)

[0091] and

Re{I _(n)}

cos(2(ω_(n) LT+Ψ _(n)))·cos Φ_(n)((p+L)T);  (19)

Im{I _(n)}

−sin(2(ω_(n) LT+Ψ _(n)))·cos Φ_(n)((p+L)T).  (20)

[0092] The FFT output in each frequency bin is a single complex numberthat resulted from M time domain inputs. Thus, the sin Φ_(n)((p+L)T) andcos Φ_(n)((p+L)T) parts of equations (17), (18), (19), and (20) that arerepresented by the real and imaginary parts of the single complex numberare an average number produced at the FFT computation rate, not theactual time domain sample rate. That is, the rate of usable Φ_(n)outputs is F_(s)/M. To achieve the desired output rate and final dataresolution, F_(s) and M are chosen such that Φ_(n)(t) has a smallvariation over the interval of M input samples. The terms with the sineand cosine of (ω_(n)LT+Ψ_(n)) in equations (17), (18), (19), and (20)are constants and are invariant in time as long as all sampling clocksand carriers are synchronized, for example, by the use of a phase lockedloop from a master clock.

[0093] In order to represent the output of the FFT more convenientlythan with equations (17), (18), (19), and (20), the following notationsthat represent the output data at the F_(s)/M output rate rather than atthe time domain input rate will be utilized.

[0094] T_(IQ)=1.0/(F_(s)/M)=output sample data interval;

[0095] F_(IQ)=1.0/T_(IQ)=output sample data rate;

[0096] a_(nr)=real part of the bin corresponding to Q_(n) at the r^(th)output time;

[0097] b_(nr)=imaginary part of the bin corresponding Q_(n) at ther^(th) output time;

[0098] c_(nr)=real part of the bin corresponding to I_(n) at the r^(th)output time; and

[0099] d_(nr)=imaginary part of the bin corresponding to I_(n) at ther^(th) output time.

[0100] The equations for a, b, c, and d for the n^(th) carrier at ther^(th) output time become:

a _(nr)=cos(ω_(n) LT+Ψ _(n))·sin Φ_(nr),  (21)

b _(nr)=−sin(ω_(n) LT+Ψ _(n))·sin Φ_(nr),  (22)

[0101] and

c _(nr)=cos(2(ω_(n) LT+Ψ _(n)))·cos Φ_(nr),  (23)

d _(nr)=−sin(2(ω_(n) LT+Ψ _(n)))·cos Φ_(nr).  (24)

[0102] The magnitude of I and Q for then^(th) carrier at the r^(th)output time may then be expressed as: $\begin{matrix}{{Q_{nr}} = {{{\sin \quad \Phi_{nr}}} = \sqrt{a_{nr}^{2} + b_{nr}^{2}}}} & (25)\end{matrix}$

[0103] and $\begin{matrix}{{l_{nr}} = {{{\cos \quad \Phi_{nr}}} = {\sqrt{c_{nr}^{2} + d_{nr}^{2}}.}}} & (26)\end{matrix}$

[0104] At step 211, signs (+ or −) are established for both I_(n)(t) andQ_(n)(t). Because equations (25) and (26) determine magnitudes for Q andI, a sign needs to be established for Q and I prior to inputting Q and Ito the arctangent function. There are many ways to establish thesesigns.

[0105] One method is to arbitrarily pick either Sign(a), i.e., the signof a, or Sign(b) for the sign of Q, and likewise either Sign(c) orSign(d) for the sign of I. As long as consistency is maintained fromoutput to output, this method may achieve reasonable results. Thismethod, however, has several drawbacks. Depending on the values of L andΨ_(n), the term picked to determine the sign may be very small and infact may be zero. In that case an instability in the sign selectionprocess may cause results to be poor in that the proper selection ofquadrant by the arctangent would be unstable. A correction for thisproblem is to adjust the start sample index, L, and Ψ_(n) such that theyare zero. Shifting L, which represents the start sample index, into theFFT is relatively easy. This process may be performed by simply throwinginput samples away (corresponding to sliding forward in time) prior toloading the FFT input buffer. Controlling the value of Ψ_(n) may be moredifficult. Ψ_(n) is generally the result of path length differences dueto manufacturing tolerances or cable length mismatches in the sensorsystem that may be difficult or costly to control or correct.

[0106] The following describes a more robust method of establishingsigns for Q and I. The sign of the term (a or b for Q; c or d for I)having the highest energy or power content is the sign that is used forsign selection. The method maybe represented as:

if a _(nr) ² ≧b _(nr) ², then Sign(Q _(nr))=Sign(a _(nr)), else Sign(Q_(nr))=Sign(b _(nr)),  (27)

[0107] and

if c _(nr) ² ≧d _(nr) ², then Sign(I _(nr))=Sign(c _(nr)), else Sign(I_(nr))=Sign(d _(nr)).  (28)

[0108] In other words, the sign chosen for Q and I is the sign of theelement having the largest magnitude. This method may not besufficiently robust, and may result in the wrong sign selection in somesituations, such as when spurious noise is present.

[0109] A more stable method sums the power over many samples to minimizethe effects of noisy data. This method utilizes the followingsummations: $\begin{matrix}{{S_{na} = {\sum\limits_{r = 1}^{W}\quad a_{nr}^{2}}},} & (29) \\{{S_{nb} = {\sum\limits_{r = 1}^{W}\quad b_{nr}^{2}}},} & (30) \\{{S_{nc} = {\sum\limits_{r = 1}^{W}\quad c_{nr}^{2}}},} & (31) \\{{S_{nd} = {\sum\limits_{r = 1}^{W}\quad d_{nr}^{2}}},} & (32)\end{matrix}$

[0110] where W=the number of output samples to sum. A good example of anumber for W is 100. The S terms from equations (29), (30), (31), and(32) are used in place of the individual terms in logical equations (27)and (28), yielding:

if S _(na) ≧S _(nb), then Sign(Q _(nr))=Sign(a _(nr)), else Sign(Q_(nr))=Sign(b _(nr)),  (33)

[0111] and

if S _(nc) ≧S _(nd), then Sign(I _(nr))=Sign(c _(nr)), else Sign(I_(nr))=Sign(d _(nr)).  (34)

[0112] This summation process to determine the S terms above need not beperformed continually. The summations may be performed at system startupand/or during any calibration cycle when only the carriers are on withor without any Φ_(n)(t) signals present. Because the phasors in eachfrequency bin effectively do not rotate, the same selected terms at eachr^(th) FFT calculation are always used to put the sign on the magnitudeterms. If the selected term has a negative sign, the associatedmagnitude term is assigned a negative sign. If the selected term has apositive sign, the associated magnitude term is assigned a positivesign. This summation process may take place in the calibration path 125.

[0113] This technique may occasionally yield a result with apositive-negative ambiguity in the final Φ_(n)(t) data. The data maycome out continuously with a sign opposite to that of the true sign. Inmany cases, such a result is not a problem. When the data of interest isrelative data, the sign is not as critical as is consistency of the signfrom output sample to output sample from a given carrier. A problem mayarise, however, if consistency across multiple carriers on a commonconductor or across multiple conductors is required. This characteristicis particularly important if the final data is to be input to a spatialbeam-former or other similar device in the downstream processing.

[0114] The above problem largely results from the fact that L andparticularly Ψ_(n) are usually unknown. While L is the same for allcarriers, Ψ_(n) is unique to each carrier. For example, suppose that thesign of a_(nr) is used for the sign of Q_(nr). In equation (21),cos(ω_(n)LT+Ψ_(n)) may start out as positive or negative depending onthe arguments. Because L and Ψ_(n) are unknown, an ambiguity may arise.The variable L may be adjusted by sliding the FFT forward, but the Ψ_(n)for each carrier is more difficult to determine. Rather than botheringto determine either L or Ψ_(n), the following calibration technique maybe utilized to resolve the problem.

[0115] By transmitting the same calibration signal Φ_(n)(t)=Φ(t) for alln, i.e., on each carrier, in the system at startup, a sign correctionvector may be determined, thereby eliminating the above problem. Eachterm of the sign correction vector is either +1 or −1, and is associatedwith a specific carrier. The sign correction vector is applied to eachcarrier's Φ_(n)(t) output to either invert or not invert the sign duringoutput processing. In this manner, coherence across carriers may bemaintained. The calibration cycle may be repeated as necessary, forexample, when the sample clock or carriers lose synchronization or whenthe system is restarted. This calibration process may take place in thecalibration path 125. This problem is not unique to the FFT approach,and is present in the homodyne approach as well. Thus, the abovecalibration technique may be applied to each of the above methods ofdetermining signs, and may be applied in a similar way in the homodyneapproach.

[0116] At step 213, the arctangent of Q_(n)(t)/I_(n)(t) is obtained,yielding Φ_(n)(t). This process, and subsequent processing steps, aresimilar for both FFT and homodyne approaches. If at step 215 there aremore carriers to be processed, the process continues with step 207,otherwise the process ends.

[0117] The present invention provides a method and apparatus fordemodulating multiple-carrier phase-modulated signals without producingunwanted signals that need to be filtered out. The present inventiondirectly converts data, utilizing a Fourier transform, without having tofirst spectrally shifting data to DC. When an FFT is utilized as theFourier transform, the benefits of reduced computation are realized,resulting in a receiver that has a reduced power consumption and fits ina smaller space when compared to previous methods.

[0118] The present invention may be embodied in other specific formswithout departing from its spirit or essential characteristics. Thedescribed embodiments are to be considered in all respects only asillustrative and not restrictive. The scope of the invention is,therefore, indicated by the appended claims rather than by the foregoingdescription. All changes that come within the meaning and range ofequivalency of the claims are to be embraced within their scope.

What is claimed is:
 1. A method comprising the steps of: receiving amultiple-carrier phase-modulated signal; converting the multiple-carrierphase-modulated signal from analog to digital, yielding a digitalsignal; performing a Fourier transform on the digital signal, yielding aplurality of real and imaginary elements associated with an n^(th)carrier, where n is an integer; determining a magnitude of an in-phasecomponent for the n^(th) carrier and a magnitude of a quadraturecomponent for the n^(th) carrier from the plurality of real andimaginary elements associated with the n^(th) carrier; establishing asign for the in-phase component for the n ^(th) carrier from a firstsubset of the plurality of real and imaginary elements associated withthe n^(th) carrier; establishing a sign for the quadrature component forthe n^(th) carrier from a second subset of the plurality of real andimaginary elements associated with the n^(th) carrier; processing thein-phase component and the quadrature component to yield an outputsignal associated with the n^(th) carrier.
 2. The method of claim 1,wherein the plurality of phase-modulated carrier signals are opticalsignals, and wherein the method further comprises the step of convertingthe optical signals to electrical signals after the receiving step. 3.The method of claim 1, further comprising the step of anti-aliasfiltering the plurality of phase-modulated carrier signals after thereceiving step.
 4. The method of claim 1, wherein the Fourier transformis a discrete Fourier transform.
 5. The method of claim 1, wherein theFourier transform is a fast Fourier transform.
 6. The method of claim ifwherein the first subset of the plurality of real and imaginary elementsassociated with the n^(th) carrier is associated with the secondharmonic of the n^(th) carrier.
 7. The method of claim 1, wherein themagnitude of the quadrature component is determined from a magnitude ofat least one real and at least one imaginary element associated with then^(th) carrier and wherein the magnitude of the in-phase component isdetermined from a magnitude of at least one real and at least oneimaginary element associated with a second harmonic of the n^(th)carrier.
 8. The method of claim 1, wherein the magnitude of thequadrature component at an r^(th) output time is |Q _(nr) |={squareroot}{square root over (a_(nr) ²+b_(nr) ²)} and the magnitude of thein-phase component at the r^(th) output time is |I _(nr) |={squareroot}{square root over (c_(nr) ²+d_(nr) ²)}, where: r is an integer;a_(nr) represents a real part of the second subset of the plurality ofreal and imaginary elements associated with the n^(th) carrier andcorresponding to Q_(n) at the r^(th) output time; b_(nr) represents animaginary part of the second subset of the plurality of real andimaginary elements associated with the n^(th) carrier and correspondingQ^(n) at the r^(th) output time; c_(nr) represents a real part of thefirst subset of the plurality of real and imaginary elements associatedwith the n^(th) carrier and corresponding to I_(n) at the r^(th) outputtime; and d_(nr) represents an imaginary part of the first subset of theplurality of real and imaginary elements associated with the n_(th)carrier and corresponding to I_(n) at the r^(th) output time.
 9. Themethod of claim 1, further comprising the steps of: transmitting acalibration signal on each carrier; determining an n^(th) signcorrection vector for the n^(th) carrier; applying the n^(th) signcorrection vector to the n^(th) carrier's output signal.
 10. The methodof claim 1, wherein the step of establishing the sign for the in-phasecomponent comprises selecting as the sign for the in-phase component oneof a sign of a real element associated with a second harmonic of then^(th) carrier and a sign of an imaginary element associated with asecond harmonic of the n^(th) carrier.
 11. The method of claim 1,wherein the step of establishing the sign for the quadrature componentcomprises selecting as the sign for the quadrature component one of asign of a real element associated with a first harmonic of the n^(th)carrier and a sign of an imaginary element associated with a firstharmonic of the n^(th) carrier.
 12. The method of claim 1, wherein thestep of establishing the sign for the in-phase, I, component comprises:if c _(nr) ² ≧d _(nr) ², then Sign(I _(nr))=Sign(c _(nr))=Sign(I_(nr))=Sign(d _(nr)), where: r is an integer corresponding to an outputtime; c_(nr) represents a real part of the first subset of the pluralityof real and imaginary elements associated with a second harmonic of then^(th) carrier; and d_(nr) represents an imaginary part of the firstsubset of the plurality of real and imaginary elements associated withthe second harmonic of the n^(th) carrier.
 13. The method of claim 1,wherein the step of establishing the sign for the quadrature component,Q, comprises: if a _(nr) ² ≧b _(nr) ², then Sign(Q _(nr))=Sign(a _(nr)),else Sign(Q _(nr))=Sign(b _(nr)), where: r is an integer correspondingto an output time; a_(nr) represents a real part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier; and b_(nr) represents an imaginary part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier.
 14. The method of claim 1, wherein the step of establishing thesign for the in-phase component, I, comprises: if S _(nc) ≧S _(nd), thenSign(I _(nr))=Sign(c _(nr)), else Sign(I _(nr))=Sign(d _(nr)), where:$\begin{matrix}{{S_{nc} = {\sum\limits_{r = 1}^{W}\quad c_{nr}^{2}}},} \\{{S_{nd} = {\sum\limits_{r = 1}^{W}\quad d_{nr}^{2}}},}\end{matrix}$

r is an integer corresponding to an output time; W=the number of outputsamples to sum; c_(nr) represents a real part of the first subset of theplurality of real and imaginary elements associated with a secondharmonic of the n^(th) carrier; and d_(nr) represents an imaginary partof the first subset of the plurality of real and imaginary elementsassociated with the second harmonic of the n^(th) carrier.
 15. Themethod of claim 1, wherein the step of establishing the sign for thequadrature component, Q, comprises: if S _(na) ≧S _(nb), then Sign(Q_(nr))=Sign(a _(nr)), else Sign(Q _(nr))=Sign(b _(nr)), where:$\begin{matrix}{{S_{na} = {\sum\limits_{r = 1}^{W}\quad a_{nr}^{2}}},} \\{{S_{nb} = {\sum\limits_{r = 1}^{W}\quad b_{nr}^{2}}},}\end{matrix}$

r is an integer corresponding to an output time; W=the number of outputsamples to sum; a_(nr) represents a real part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier; and b_(nr) represents an imaginary part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier.
 16. An apparatus comprising: an analog-to-digital converter,arranged and constructed to convert a multiple-carrier phase-modulatedsignal into a digital signal; a Fourier transformer, arranged andconstructed to perform a Fourier transform on the digital signal,yielding a plurality of real and imaginary elements associated with ann^(th) carrier, where n is an integer; a magnitude determiner, arrangedand constructed to determine a magnitude of an in-phase component forthe n^(th) carrier and a magnitude of a quadrature component for then^(th) carrier from the plurality of real and imaginary elementsassociated with the n^(th) carrier; an in-phase sign establisher,arranged and constructed to establish a sign for the in-phase componentfor the n^(th) carrier from a first subset of the plurality of real andimaginary elements associated with a second harmonic of the n^(th)carrier; a quadrature sign establisher, arranged and constructed toestablish a sign for the quadrature component for the n^(th) carrierfrom a second subset of the plurality of real and imaginary elementsassociated with the n^(th) carrier; a processor, arranged andconstructed to process the in-phase component and the quadraturecomponent to yield an output signal associated with the n^(th) carrier.17. The apparatus of claim 16, further comprising a calibration path,arranged and constructed to determine an n^(th) sign correction vectorfor the n^(th) carrier from a calibration signal transmitted on eachcarrier and apply the n^(th) sign correction vector to the n^(th)carrier's output signal.
 18. The apparatus of claim 16, wherein themagnitude determiner is further arranged and constructed to determinethe magnitude of the quadrature component from a magnitude of at leastone real and at least one imaginary element associated with the n^(th)carrier and to determine the magnitude of the in-phase component from amagnitude of at least one real and at least one imaginary elementassociated with a second harmonic of the n^(th) carrier.
 19. Theapparatus of claim 16, wherein the in-phase sign establisher is furtherarranged and constructed to establish the in-phase sign based on atleast one real element and at least one imaginary element associatedwith a second harmonic of the n^(th) carrier; and wherein the quadraturesign establisher is further arranged and constructed to establish thequadrature sign based on at least one real element and at least oneimaginary element associated with a first harmonic of the n^(th)carrier.
 20. The apparatus of claim 16, wherein the in-phase signestablisher is further arranged and constructed to establish the signfor the in-phase component, I, by performing: if S _(nc) ≧S _(nd), thenSign(I _(nr))=Sign(c _(nr)), else Sign(I_(nr))=Sign(d _(nr)), where:$\begin{matrix}{{S_{nc} = {\sum\limits_{r = 1}^{W}\quad c_{nr}^{2}}},} \\{{S_{nd} = {\sum\limits_{r = 1}^{W}\quad d_{nr}^{2}}},}\end{matrix}$

r is an integer corresponding to an output time; W=the number of outputsamples to sum; c_(nr) represents a real part of the first subset of theplurality of real and imaginary elements associated with a secondharmonic of the n^(th) carrier; and d_(nr) represents an imaginary partof the first subset of the plurality of real and imaginary elementsassociated with the second harmonic of the n^(th) carrier; and whereinthe quadrature sign establisher is further arranged and constructed toestablish the sign for the quadrature component, Q, by performing: if S_(na) ≧S _(nb), then Sign(Q _(nr))=Sign(a _(nr)), else Sign(Q_(nr))=Sign(b _(nr)), where: $\begin{matrix}{{S_{na} = {\sum\limits_{r = 1}^{W}\quad a_{nr}^{2}}},} \\{{S_{nb} = {\sum\limits_{r = 1}^{W}\quad b_{nr}^{2}}},}\end{matrix}$

r is an integer corresponding to an output time; W=the number of outputsamples to sum; a_(nr) represents a real part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier; and b_(nr) represents an imaginary part of the second subset ofthe plurality of real and imaginary elements associated with the n^(th)carrier.